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1. Show that m*(Q) = 0, where Q is the set of rational numbers. 2. Let x e R. Then m*(A) = m*(x + A) for all A c R. 3. Find the length of the set U{x:

1. Show that m*(Q) = 0, where Q is the set of rational numbers. 2. Let x e R. Then m*(A) = m*(x + A) for all A c R. 3. Find the length of the set U{x:

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Show that m*(Q) = 0, where Q is the set of rational numbers. 2. Let x e R. Then m*(A) = m*(x + A) for all A c R. 3. Find the length of the set U {:< < 4. Show that if m* (E) = 0, then E is measurable. 5. Prove that if m* (E,) = 0, then m*(E, U E2) = m*(E2). 6. Prove that if O is any open set, then 0 is measurable and m(0) = l(0). 7. Let f be a real-valued function and a be a o - algebra on R. Then f is measurable if and only if f-'(B) is measurable
Transcribed Image Text:1. Show that m*(Q) = 0, where Q is the set of rational numbers. 2. Let x e R. Then m*(A) = m*(x + A) for all A c R. 3. Find the length of the set U {:< < 4. Show that if m* (E) = 0, then E is measurable. 5. Prove that if m* (E,) = 0, then m*(E, U E2) = m*(E2). 6. Prove that if O is any open set, then 0 is measurable and m(0) = l(0). 7. Let f be a real-valued function and a be a o - algebra on R. Then f is measurable if and only if f-'(B) is measurable
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